us federal reserve: 200332pap
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Ito Conditional Moment Generator andthe Estimation of Short Rate Processes1
Hao Zhou
Mail Stop 91Federal Reserve BoardWashington, DC 20551
First Draft: January 1998This Version: March 2003
1This paper grows out of an essay of my PhD dissertation, and is previously distributed under thetitle “Jump-Diffusion Term Structure and Ito Conditional Moment Generator.” George Tauchengave me valuable advice. I thank Ravi Bansal, David Bates, Tim Bollerslev, Peter Christoffersen,Sanjiv Das, Lars Hansen, Michael Hemler, Nour Meddahi, and Kenneth Singleton for their helpfulsuggestions. I am also grateful to the editor Rene Garcia, the associate editor, and two anonymousreferees for their constructive suggestions. Comments from the seminar participants at Duke,Brown, Virginia economics departments, SNDE 1999, FMA 1999, ES 2000 annual meetings, andDuke Risk Conference 2000 are greatly appreciated. The views presented here are solely thoseof the author and do not necessarily represent those of the Federal Reserve Board or its staff.For questions and comments, please contact Hao Zhou, Trading Risk Analysis Section, Division of Research and Statistics, Federal Reserve Board, Washington DC 20551 USA; Phone 1-202-452-3360;Fax 1-202-728-5887; e-mail [email protected].
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Abstract
This paper exploits the Ito’s formula to derive the conditional moments vector for the class
of interest rate models that allow for nonlinear volatility and flexible jump specifications.
Such a characterization of continuous-time processes by the Ito Conditional Moment Gener-
ator noticeably enlarges the admissible set beyond the affine jump-diffusion class. A simple
GMM estimator can be constructed based on the analytical solution to the lower order mo-
ments, with natural diagnostics of the conditional mean, variance, skewness, and kurtosis.
Monte Carlo evidence suggests that the proposed estimator has desirable finite sample prop-
erties, relative to the asymptotically efficient MLE. The empirical application singles out the
nonlinear quadratic variance as the key feature of the U.S. short rate dynamics.
Keywords: Ito Conditional Moment Generator, Short Term Interest Rate, Jump-Diffusion
Process, Quadratic Variance, Generalized Method of Moments, Monte Carlo Study.
JEL classification: C51, C52, G12
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1 Introduction
In modeling the short-term interest rate, researchers face the challenge of accommodating all
relevant features in a single model specification. Those features, include but are not limitedto, (1) short-term persistence, (2) long-run mean reversion, (3) nonlinear state-dependence
in volatility, (4) non-Gaussian features in skewness and kurtosis. The celebrated CIR model
(Cox et al., 1985) and its various extensions, although appealing in their general equilibrium
nature and closed-form solution, have difficulty in fitting all these features simultaneously
for the US interest rate data (Brown and Dybvig, 1986).1 Rigorous specification tests using
historical data tend to reject the square-root model (Aıt-Sahalia, 1996b; Conley et al., 1997;
Gallant and Tauchen, 1998). Although having an inherent advantage in fitting features (1)
and (2) as indicated in the literature, the CIR-type model fails to capture the rich volatility
dynamics and the nonlinear non-Gaussian features.
Consequently, efforts to modify the square-root model largely concentrate on more flexible
specifications of the volatility dynamics. It is clear that the CIR model is just one special case
of so-called linear CEV (constant elasticity of volatility) specification, where the elasticity
equals one half. Recent comparative studies (Chan et al., 1992; Conley et al., 1997; Tauchen,
1997; Christoffersen and Diebold, 2000) found that an elasticity around one and one-half is
more desirable. Alternatively, one can estimate the volatility function nonparametrically
(Aıt-Sahalia, 1996a; Stanton, 1997; Jiang and Knight, 1997; Jiang, 1998; Bandi, 2002; Bandi
and Phillips, 2003). The empirical findings along this line suggest that the square-root model
fits reasonably well for the medium range of interest rates, but the estimated nonlinearity
at both the high and low ends is neither accurate nor conclusive. A pertinent approach is to
introduce an unobserved stochastic volatility factor into the diffusion function, which finds
considerable support in empirical studies (Andersen and Lund, 1996, 1997).2 The jump-
diffusion approach to interest-rate modeling (and bond pricing exercise) is of more recent
1
The bivariate extensions of CIR specification (Gibbons and Ramaswamy, 1993; Chen and Scott, 1993;Pearson and Sun, 1994) also meet with poor empirical performance. Duffie and Singleton (1997) foundfavorable evidence for a two-factor CIR model with serially-correlated error structure. Dai and Singleton(2000) estimated more flexible three-factor affine specifications similar to Chen (1996) and Balduzzi et al.(1996) for interest rate swap data after 1987.
2This paper focuses on the maximum flexibility in the univariate setting, and the extension to multivariateor stochastic volatility is deferred to future research.
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origin (Baz and Das, 1996; Das, 1998), and its general equilibrium formulation is explored
by Ahn and Thompson (1988).3
The innovation of this paper is to generate the parametric conditional moments only using
the Ito’s formula and to construct a computationally efficient GMM estimator. Maximum
Likelihood Estimation (MLE) is available only for a very restricted class of jump-diffusion
models (Lo, 1988). Our method differs with the infinitesimal generator of Hansen and
Scheinkman (1995) (GMM) in that it fully exploits the conditional information, does not
rely on simulations as do Duffie and Singleton (1993) (SMM), uses model-dependent moments
instead of data-dependent moments (Gallant and Tauchen, 1996) (EMM), generalizes to an
arbitrary number of moments rather than only to conditional mean and variance (Fisher
and Gilles, 1996) (QML), and has reliable small sample properties in comparison with the
nonparametric approach (Aıt-Sahalia, 1996a) (NP). As shown below, our method reduces
a complicated task of solving a stochastic differential equation (SDE) to a simple matrix
solution of an ordinary differential equation (ODE) system. The computational burden is
reduced to a minimum of elementary algebra.4 Another important advantage is that the
characterization of short rate processes by the Ito’s approach allows for nonlinear volatility
and semiparametric jump specifications. Within the univariate paradigm, nonlinearity is
indispensable to successful modeling of the U.S. short term interest rate. In the literature,
the most closely related method is to identify the stochastic differential equations with anorthogonal series representation (Hansen et al., 1998), which is attributed to the generalized
eigenvalue-eigenfunction technique (Wong, 1964).
We further justify the aforementioned methodology by a numerical exercise and illustrate
by an empirical application. Monte Carlo evidence suggests that the finite sample efficiency
of the proposed GMM estimator is comparable to the asymptotically efficient MLE, the
3Recently there is a growing literature on jump-diffusion interest rate modeling (see Chacko and Das,1999; Johannes, 1999; Piazzesi, 2000, among many others), which ranges from short-rate dynamics to fixed-income derivatives, from market-implied jumps to macroeconomic announcements, and from parametric to
nonparametric specifications.4Alternatively, an equivalent spectral method of moments is developed by exploiting the closed-form
conditional characteristic functions for the affine jump-diffusion model (Chacko and Viceira, 1999; Singleton,2001; Jiang and Knight, 2002; Carrasco et al., 2002). However, the selection of spectral moments remains as adifficult challenge, whereas in the classical method of moments, a natural choice is the lower-order moments.Moreover, a strategy to derive moments using the Ito’s formula alone and not relying on the characteristicfunction or moment generating function may be more desirable for certain non-standard processes, e.g., thequadratic variance model discussed in this paper.
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sampling t-statistics of individual parameter is not far away from the normal reference dis-
tribution, and the GMM test of over-identifying restrictions has a typical upward bias but
with reasonable magnitude. When applied to the U.S. short term interest rate from 1954
to 2002, both the square-root model and the restricted CEV model are rejected outright.
Adding jumps shows some improvement but only the quadratic variance model cannot be
rejected at the 1% significance level. U-shaped volatility and nonlinear higher order moments
seem to be the main challenges of fitting the U.S. short rate dynamics, in additional to the
well-known linear mean persistence.
The remainder of this paper is organized as follows: Section 2 derives the conditional
moments for an admissible class of processes including the square-root, the restricted CEV,
the jump-diffusion, and the quadratic variance; Section 3 builds an easy-to-implement GMM
estimator and provides some finite sample evidence; Section 4 applies the estimating proce-
dure to the four models mentioned above and contrasts the specification differences using
the conditional moment profiles; and Section 5 concludes.
2 Ito Conditional Moment Generator
This section outlines a strategy to derive the conditional moments simultaneously for certain
continuous-time processes, relying only on the Ito’s formula and the specifications of drift,
diffusion, and jump functions. The resulting characterization not only nests the popular
affine jump-diffusion class, but also features nonlinear quadratic variance and semiparametric
flexible jumps.
2.1 A General Characterization of Admissible Processes
Suppose that the evolution of the state variable (i.e., the short rate) is governed by a reduced-
form jump-diffusion process
drt = µtdt + σtdW t + J tdN (ρtt), (1)
where W t is a standard Brownian motion, N (ρtt) is a Poisson driving process with an inten-
sity function ρt, and J t is the jump size with distribution Π(J t). Note that both the jump
rate and jump size are allowed to be state-dependent but conditionally independent of each
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other and with respect to the Brownian motion. Process (1) must satisfy certain regularity
conditions and the critical ones are: (a) both µt and σt are Lipschitz continuous, (b) ρt and
Π(J t) are
F t−
measurable.
The strategy is to solve all the conditional moments up to the K ’th order simultaneously,
by first applying the Generalized Ito’s lemma (Merton, 1971; Lo, 1988) to each rkT for k =
1, 2, · · · , K , and then take the conditional expectation
E t(rkT ) = rkt + E t
T t
µukrk−1u +
1
2σ2uk(k − 1)rk−2u + ρuE J [(ru + J u)k − rku]
du
(2)
Interchanging the expectation and integration operators, and taking the derivative with
respect to time T , we arrive at a differential equation system
dE t(rks
)
ds = E t
µskrk−
1s +1
2σ2sk(k − 1)rk−
2s + ρsk
i=1
ki
rk−
is E J (J is)
. (3)
with boundary condition E t(rkt ) = rkt . The following proposition characterizes the class of
jump-diffusion processes that sustain a closed-form solution to equation (3),
Proposition 1 (Characterization) The sufficient condition for the K-dimensional ordi-
nary differential equation system (3) to have a first order linear solution, is to restrict the
drift, diffusion, and jump functions in the following forms
(1) µt = κ(θ
−rt); and
(2) σt =
σ0 + σ1rt + σ2r2t ; and
(3) ρtE J (J kt ) =k
j=0 J kjr jt .
Many linear or nonlinear restrictions need to be imposed to ensure existence and identifi-
cation, for example, the sign constraints on κ,θ,σ0, σ1, σ2 and the zero constraints on some
J kj . The proof only involves a straightforward verification, hence omitted.5
For the admissible process under Proposition 1, the K -vector of its conditional moments
E t(Rs) = [E t(rs), E t(r2s), · · · , E t(rK s )]
is characterized by a linear differential equation sys-tem,
dE t(Rs)
ds= A(β )E t(Rs) + g(β ), (4)
5The necessary conditional for the K -dimensional ordinary differential equation system (3) to have a first
order linear solution, is to require the termµskr
k−1s + 1
2σ2sk(k − 1)rk−2s + ρs
k
i=1
ki
rk−is
E J (J is)
to be a
k’th order polynomial of rs, which is trivial and not as informative as the sufficient condition.
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where A(·) is a K × K lower-triangular matrix and g(·) is a K × 1 vector. Both A(·)and g(·) are nonlinear functions of the parameter vector β = [κ,θ,σ0, σ1, σ2, J 10, · · · , J KK ]
,
defined by the underlying the jump-diffusion process (1). Since the coefficients of such a
non-homogeneous linear first-order differential equation do not depend on time, one obtains
the following closed-form solution,
E t(RT ) = e(T −t)A(β )Rt + A−1(β )
e(T −t)A(β ) − I
g(β ), (5)
where I is the K × K identity matrix and e(·) denotes the matrix exponential.
There are some advantages in using the “Ito Transformation” to generate the conditional
moments. From the perspective of richer dynamics, although the drift function has to be
restricted as linear, the diffusion function can be nonlinear, and the jump function only re-quires the specification of its moments. More detailed examples are examined in the next
subsection to illustrate the enhanced flexibility of such an Ito characterization. From the
perspective of easier implementation, the calculation of moments in a typical matrix pro-
gramming language remains a one-line code as equation (5), and the computation of each
entry of A(·) and g(·) in equation (3) does not require differentiation; whereas using the
conditional moment generating function involves messy high order derivatives. Once com-
puted, a moment-based estimator (like GMM) is readily available, while a likelihood-based
method requires the Fourier inversion of the characteristic function. It is also possible to
apply the Ito transformation to processes that lack analytical solution to the moment gener-
ating function. The major disadvantage of relying on a potentially limited set of moments, is
the possible loss of estimation efficiency relative to MLE. To address this concern, the next
section designs a GMM estimator and quantifies its adequate finite sample performance.
2.2 Leading Empirical Examples
To illustrate the applicability of the proposed methodology, here we present several specifi-
cations that are useful to model the short term interest rate. Only the solutions to the first
four moments are spelled out, as the higher order moments are trivial extensions.
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2.2.1 Flexible Jump-Diffusion Process
We start with a simple jump-diffusion process
drt = κ(θ − rt)dt + σ√rtdW t + J tdN (ρtt) (6)
where ρt = ρ and J t is specified by its four moments. Although the diffusion part of this
model is affine, the state variable may not be affine if the jump-size moments are state-
dependent as in Proposition 1. The solution to its first four conditional moments in the form
of equation (5), can be characterized by the matrix A(β )
−κ + ρE (J ) 0 0 0
2κθ + σ2 −2κ + ρ2
i=1 (2i ) E (J i) 0 0
0 3κθ + 3σ2
−3κ + ρ3
i=1 (3i ) E (J i) 0
0 0 4κθ + 6σ2 −4κ + ρ4
i=1 (4i ) E (J i)
and the vector g(β )
κθ
0
0
0
If we specialize to the case of uniform distribution (J t ∼ U[art, brt]), the moments of the
jump-size are, respectively, E (J ) = (b2−a2)2(b−a)
rt, E (J 2) = (b3−a3)3(b−a)
r2t , E (J 3) = (b4−a4)4(b−a)
r3t , and
E (J 4) = (b5−a
5
)5(b−a) r4t .
Two important points are worth noting here. First, the model is not affine as the con-
ditional variance is not linear but quadratic in the state variable, which is qualitatively
similar to the quadratic variance diffusion model discussed next. Second, the particular
state-dependence of the jump-size rules out the possibility of negative interest rate, under
the mild restriction that −1 ≤ a < b < +∞. Negative short rate level is difficult to dealt
with for certain affine specifications and is conceptually problematic in a nominal economic
environment.
2.2.2 Quadratic Variance Diffusion Model
An important alternative to the affine variance model is the “quadratic variance” process
defined as
drt = κ(θ − rt)dt +
σ20 − σ2
1rt + σ22r2t dW t (7)
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No sign restrictions are imposed in the GMM estimation procedure, but are adopted here in
line with the actual result to highlight some nice properties—non-zero volatility when rate
approaches zero, high volatility when rate is high, and comparable scale of the local variance
parameter (as in the square-root model).
For this quadratic variance model, the conditional moments are characterized by the
equation (5) in terms of
A(β ) =
−κ 0 0 0
2κθ − σ21 −2κ + σ2
2 0 0
3σ20 3κθ − 3σ2
1 −3κ + 3σ22 0
0 6σ20 4κθ − 6σ2
1 −4κ + 6σ22
and
g(β ) =
κθ
σ20
0
0
Note that the solution structure is similar for both the jump-diffusion and the quadratic
variance models, and that the only difference is in each entry. This feature makes the
numerical calculation of the moments straightforward and fast.
The quadratic variance model has several important advantages. First, the model is notaffine hence its moment generating function or characteristic function may not be easy to
derive. Then the Ito conditional moment generator may be the only choice among all the
non-simulation-based methods to calculate the moments. Second, there is a great deal of
debate about whether the volatility is linear or nonlinear, e.g., the “U” shaped volatility
pattern reported by Aıt-Sahalia (1996a). Here we can provide a simple parametric nonlinear
alternative and a feasible GMM estimator with conditional moment based diagnostics. Third,
the quadratic variance specification seems to nest several famous short rate models, namely,
log-linear (σ0 = σ1 = 0), Ornstein-Uhlenbeck (σ1 = σ2 = 0), and square-root (σ0 = σ2 = 0
and reversing the sign of σ21). Of course, the obvious disadvantage is that the bond pricing
solution is not easily obtained except for using Monte Carlo simulation. Nevertheless, the
empirical evidence of Section 4 seems to suggest that the quadratic variance function is
indispensable in modeling the univariate short rate dynamics.
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2.2.3 Cubic or Transformable CEV Model
Some models are not directly solvable by the Ito conditional moment generator, but can be
“reduced” to the tractable cases by appropriate transformations. For a detailed discussionon the reducibility technique, see Chapter 4 of Kloeden and Platen (1992). Consider the
following nonlinear drift and constant-elasticity-of-volatility (CEV) specification
drt = κ(θr2γ −1t − rt)dt + σrγ t dW t, (8)
where under appropriate parameter restrictions rt ∈ (0, +∞) and has a positive starting
value. Note that the cross restriction on parameter γ between drift and diffusion is required
for the reducibility, and may prove to be empirically too restrictive relative to the standard
linear drift CEV model. Marsh and Rosenfeld (1983) first proposed such a modeling strategy
and estimated with maximum likelihood for distinct values of γ = 0, 0.5, 1. Eom (1997)
studied the distributional properties and the optimal GMM instruments for γ ∈ [0, 1). Ahn
and Gao (1999) examined the term structure implications for the case of γ = 1.5 and
estimated with GMM. The adopted GMM estimators were based on time discretization and
approximate first and second moments.
Using the transformation xt = rαt , which is a state-preserving transformation when rt ∈(0, +
∞) and γ
∈[0, 1) or γ
∈(1, +
∞), one arrives at the familiar square-root model6
dxt = a(b − xt)dt + c√
xtdW t. (9)
The above transformation can be characterized by the following proposition
Proposition 2 (Transformation) The mappings between the CEV process (8) and the
square-root model (9) are
α = 2(1 − γ )
a = 2(1− γ )κ
b = θ +(1−2γ )σ2
2κ
c = 2(1− γ )σ
(10)
The proof is a straightforward application of the Ito’s lemma, and is available from the author
upon request. The solution to conditional moment of the transformed process xt is a special6When γ = 1 the square-root process (8) is reduced to the log-normal process drt = κ(θ−1)rtdt+σrtdW t,
and the parameters κ and θ are not separately identifiable.
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case of the jump-diffusion process (6) without jumps (letting ρt = 0 would be sufficient). The
fourth parameter γ in the nonlinear drift CEV model (8) is identified through the nonlinear
but monotonic transformation xt = rαt , given that rt
∈(0, +
∞).
3 Estimation Strategy and Monte Carlo Evidence
Deriving the conditional moment restrictions (5) only achieves half of the task for estimating
the underlying continuous-time model. The other half rests on designing an appropriate
estimator with desirable large and small sample properties. The purpose of the this section
is to outline an easy-to-implement GMM estimator based on the moment condition solution
(5), and to assure the readers that the estimator performs reasonably well for the benchmark
CIR model under empirically plausible scenarios.
3.1 The GMM Estimator
The condition moments solution (5) can be spelled out as a vector-auto-regressive (VAR)
formula
E t[ht+1(β )] =
E t(rt+1)
E t(r2t+1)
E t(r3
t+1)E t(r4t+1)
−
d11 0 0 0
d21 d22 0 0
d31 d32 d33 0d41 d42 d43 d44
rt
r2t
r3
t
r4t
−
d01
d02
d03d04
= 0 (11)
which is a recursive simultaneous equation system and its unrestricted version can be esti-
mated by the ordinary least square (OLS). To form a generalized method of moments (GMM)
estimator, a natural choice of instruments is the constant one and the lagged variables, hence
the moment condition vector (with a total of fourteen equations)
f t(β ) ≡
(E (rt+1) − rt+1)(1, rt)
(E (r2t+1
)−
r2t+1
)(1, rt, r2
t)
(E (r3t+1) − r3t+1)(1, rt, r2t , r3t )
(E (r4t+1) − r4t+1)(1, rt, r2t , r3t , r4t )
(12)
By construction E [f t(β 0)] = 0, and the corresponding GMM or minimum chi-square esti-
mator is defined by β T = argmin gT (β )W gT (β ), where gT (β ) refers to the sample mean
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of the moment conditions, gT (β ) ≡ 1/T T −1
t=1 f t(β ), and W denotes the asymptotic co-
variance matrix of gT (β 0) (Hansen, 1982). An iterative estimator of W is adopted here; and
since the error is not serially correlated, only the heteroscedasticity need to be accounted for.
Under standard regularity conditions, the minimized value of the objective function (normal-
ized by the sample size) is asymptotically distributed a chi-square random variable, which
allows for an omnibus test of the overidentifying restrictions. Moreover inference regard-
ing individual parameters is readily available from the standard formula of the asymptotic
variance-covariance matrix, (∂f t(β )/∂β W ∂f t(β )/∂β )/T .
3.2 Considerations for Identification and Efficiency
Identification, or global identification, is equivalent to the assumption that the GMM estima-tor achieves a unique minimum at some β 0 ∈ B, where B is a compact set. In the unrestricted
recursive VAR model (11), the total number of identifiable parameters is fourteen, which can
be easily verified by the standard order and rank conditions. Since the underlying jump-
diffusion model is nonlinear, the identification issue becomes more complicated—on the one
hand, the restricted nonlinear dynamics may not be able to identify as many as fourteen pa-
rameters; on the other hand a nonlinear structure usually helps to identity more parameters
than a linear structure. There is not much theoretical guidance in literature on how to ver-
ify the identification condition in a nonlinear model before the model is actually estimated.
However, there is a sufficient condition—plim(∂gT /∂β W T ∂gT /∂β )/T being nonsingular—
that can be numerically verified with the estimation result from a given sample data set. It
is equivalent to the more primitive condition of local identification that the gradient is of
full column rank and the Hessian is negative definite. In practice, all the empirical examples
seem not to violate this sufficient condition, except a variation of the jump-diffusion model
where both the jump-rate and jump-size parameters are state-independent constants.
Following Hansen (1985) and Hansen et al. (1988), the conditional moment restriction
E t[ht+1(β )] = 0 indicated by equation (11) implies an efficient choice of instruments as
E t[∂ht+1(β 0)/∂β ]V art[ht+1(β 0)]−1. In theory such a choice of instruments should be ideal,
but in practice other considerations may favor the natural choice of (12). First, the optimal
instruments involve unknown true distribution parameter β 0, which has to be approximated
in the GMM estimation procedure. Second, to calculate the optimal instruments one needs
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to solve for eight lower order moments if one uses only four lower order moments in the
estimation, which is trivial analytically using the Ito approach but may be numerically
unstable for real world data sets. Further, there is a logical inconsistency—one has the
knowledge of eight order moments but does not use it in the moment condition restriction.
Meddahi and Renault (1997) proposed an interesting treatment that reduces the information
of the third and fourth conditional moments to the unconditional skewness and kurtosis,
and achieves the efficient estimates of conditional mean and variance. The GMM estimator
implemented here explicitly incorporates the conditional third and fourth moments and is
conceptually related to their efficient estimation of the first two conditional moments. The
relative efficiency of the proposed GMM estimator can be judged in a Monte Carlo setting,
against the asymptotically efficient MLE (which is theoretically superior to GMM for a given
set of moment restrictions with optimal instruments).
3.3 Monte Carlo Evidence
To assess the finite sample performance of the proposed GMM estimator, a limited Monte
Carlo study is conducted for the benchmark square-root model drt = κ(θ−rt)dt + σ√
rtdW t,
in comparisons with the MLE estimates reported by Durham and Gallant (2002). There are
six scenarios chosen in their paper, with varying degrees of persistence and volatility, and a
fixed long-run mean of 6 percent. I adopted the exact same setup with 1000 observations ineach random sample and a total of 512 Monte Carlo replications. To avoid the discretization
bias, I simulate the square-root model from the exact non-central Chi-square distribution
f (rt+∆|rt; κ,θ,σ) = ce−u−v
v
u
q
2
I q
2√
uv
, (13)
where q = 2κθ/σ2 − 1, c = 2κ/σ2(1− e−κ∆), u = crte−κ∆, v = crt+∆, and I q(·) is a modified
Bessel function of the first kind with a fractional order q (Oliver, 1972). A composite method
of generating random number (Devroye, 1986) is adopted here after transforming the above
density function into,
f (y) =∞
j=0
y j+λ−1e−y
Γ( j + λ)· u je−u
j!=
∞ j=0
Gamma(y| j + λ, 1) · Poisson( j|u) (14)
with y = v and λ = q + 1. In practice, one first draws a random number j from the
Poisson ( j|u) distribution; then draws another random number y from the Gamma (y| j +λ, 1)
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distribution; and finally calculates the target state variable rt+∆ = y/c. See Zhou (2001) for
implementation detail.
Table 1 compares the parameter estimates of the proposed GMM estimator in this paper
with those of the MLE estimator, under the six scenarios (a-f) in Durham and Gallant
(2002). In terms of bias, only the mean-reversion parameter κ has a sizable upward bias
when the persistence level is high (scenario a, b, and d)—about 10% of the parameter value
for MLE and about 20% for GMM; while for the less persistent scenarios (c, e, and f),
the bias is noticeably reduced. This is a classical case of finite sample bias in estimating
the AR(1) coefficient for near-unit-root processes. For the long-run mean parameter θ and
the local variance parameter σ, MLE has negligible positive bias and GMM has negligible
negative biases. In terms of relative efficiency, the proposed GMM estimator is remarkably
close to the asymptotically efficient MLE. The root-mean-squared-error of GMM is no more
10% larger than that of MLE for most parameters in the persistent cases (scenarios a, b,
and d), and is practically indistinguishable for most parameters in the less persistent cases
(scenarios c, e, and f ). Figure 1 reports the GMM test of the overidentifying restrictions,
which exhibits a typical over-rejection bias but with a reasonable size comparing with the
reference level. The sampling distribution of the t-test statistics is graphed in Figure 2,
and indicates that the finite sample distortion is rather small comparing with the reference
Normal(0,1) distribution.
4 Empirical Application
In this section, the Ito moment generator and the related GMM estimator are applied to the
empirical U.S. interest rate data. The weekly 3-month t-bill rate from January 1954 to July
2002, totaling 2504 observations, is obtained from the Federal Reserve Bank of St. Louis
public website. The time series plot is given in Figure 3 and the summary statistics are
reported in Table 2. The short rate exhibits the typical features found in literature—highpersistence (auto-regressive coefficients close to one), high volatility (standard deviation
277 basis points), moderately high skewness (1.14) and kurtosis (4.87). I will focus on
the estimation result of the four empirical examples (including the benchmark CIR model)
presented in Section 2, and illustrate how to use the conditional moment functions to further
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compare different model specifications.
4.1 Estimation Result
The GMM estimator designed in Section 3 is applied to the four candidate models discussed
in Section 2: square-root, restricted CEV, jump-diffusion, and quadratic variance.7 The
results are summarized in Table 3.
The standard square-root model is strongly rejected by the GMM specification test, with
a chi-square (df=11) of 49.36. The long-run mean parameter (0.0497) is about 50 basis points
lower than the sample average (0.0542), the mean-reversion parameter is very low (0.0020)
and imprecisely estimated with standard error 0.0013, and the local variance parameter is
also lower (0.0062 which implies a unconditional standard deviation 0.0219 versus the samplestandard deviation 0.0277).
The nonlinear drift CEV model is also strongly rejected with a chi-square (df=10) of
48.57. Although most parameters are accurately estimated, the model also has difficulty
in nailing down the mean reversion parameter κ (0.0017 with standard error 0.0010). The
restricted CEV model accurately estimates elasticity parameter as 0.4825 with standard error
0.0028, which confirms the empirical finding by Eom (1997). All other parameter estimates
are close to and/or slightly lower than the square-root estimates.8
The jump-diffusion model is implemented here with a constant jump-rate ρ and a uni-
form jump-size (−art, art). The symmetry restriction on jump size is to ensure identification.
The result predicts roughly two jumps per year; with a state-dependent jump-size of plus
or minus 119 basis points at the sample average (0.0542), plus or minus 13 basis points at
sample minimum (0.0058), and plus or minus 368 basis points at sample maximum (0.1676).
Such a jump pattern is more realistic than the constant jump-size distribution, and can rule
7As pointed out by a referee, one could estimate a comprehensive model nesting both time-varying jumpsand conditional quadratic variance. I found out that such a specification is not empirically identifiable bythe GMM estimator. My intuition is that the particular jump and diffusion specifications adopted here
produce the similar quadratic conditional variance. Therefore they are substituting for each other insteadof being complementary. This can be easily seen from the diagnostic conditional moment graphs in the nextsubsection.
8This result differs from the typical empirical finding for the linear drift CEV model, in that the elasticitycoefficient there is found to be in the range of 1.0-1.5 (Chan et al., 1992; Conley et al., 1997; Tauchen,1997; Christoffersen and Diebold, 2000), possibly because that the nonlinear drift CEV model imposes anunrealistic restriction across the drift and diffusion functions.
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out the negative interest rates, which is quite troublesome in a nominal economic environ-
ment. Nevertheless, the model is rejected at the p-value 0.0002, and the parameter θ is
unconvincingly large (0.0995).
The quadratic variance model performs the best, and is not rejected at the 1% significance
level (p-value 0.0121). All the parameter estimates are highly significant. The estimates of
the drift parameters fall between the square-root model (similarly the CEV model) and the
jump-diffusion model. The parameter estimates of the diffusion function guarantee that
(a) instant variance does not admit negative value, (b) minimum volatility is achieved at
a positive short rate level, and (c) volatility increases more when short rate level is high
than when short rate level is low (see the conditional moment graphs bellow). Such a result
from a parametric perspective seems to be confirm the finding of Aıt-Sahalia (1996a) from
a nonparametric perspective.
4.2 Conditional Moment Graphs
The conditional moment vector (11) not only serves as the basis for constructing a GMM
estimator, but also provides intuitive diagnostics in conditional mean, volatility, skewness,
and kurtosis. The conditional mean and conditional variance in discrete sampling intervals,
are equivalent to the drift and volatility functions in instant times for the pure diffusion
processes, but more general in covering also the jump-diffusion processes. The conditional
skewness and kurtosis provide natural assessment on how much the implied transitional
density deviates from the conditional normality. Higher order conditional moments are
especially informative about the jump impact when the time horizon is longer than zero,
but the instantaneous higher order moments cannot provide any new information than the
instantaneous drift and volatility.
Figure 4 plots the conditional mean (top panel) and conditional variance (bottom panel).
It is clear that the square-root model has the least persistence in level. Although the re-
stricted CEV model has a potential nonlinear drift, the estimated mean function is mostly
linear and close to the square-root model. On the other hand, the jump-diffusion model is the
most persistent case, suggesting an observational equivalence between occasional jumps and
near unit-root in interest rate processes. Our preferred quadratic variance model has a linear
mean function with a moderate persistence among the four models. Turning to the condi-
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tional volatility, both the square-root model and the restricted CEV model produce nearly
identical linear volatility profiles, underpinning the clear rejection by the GMM specifica-
tion tests. Jump-diffusion process provides a slightly nonlinear quadratic variance function,
due to the state-dependent jump-size specification (J t ∼ U [−art, art]) that differs from the
standard affine jump-diffusion models. Of course the most dramatic result comes from the
U-shaped quadratic variance model, which partially confirms the nonparametric finding of
the nonlinear volatility by Aıt-Sahalia (1996a) and the parametric finding of CEV elasticity
between 1.0 and 1.5 (Chan et al., 1992; Conley et al., 1997; Tauchen, 1997; Christoffersen
and Diebold, 2000). The post-war U.S. history suggests that the interest rate volatility is
certainly high when the short rate level is high, but the volatility is also none trivial when
the rate is close to zero. Therefore a nonlinear dependence of short rate volatility on its level
may be better captured by a quadratic variance model than by a standard affine model.
Figure 5 depicts the conditional skewness and kurtosis functions and offers some assess-
ment of the departures from the conditional normality. From the top panel we can see that
both the square-root and the nonlinear CEV model give a virtually same hyperbolic skew-
ness function—shooting up at the lower end and approaching zero at the higher end. The
jump-diffusion process has a similar profile but a uniformly higher skewness once the short
rate level reaches above 2 percent. The quadratic volatility model is unique in presenting a
nonlinear increasing skewness function that approaches -0.1 at the low end and +0.1 at thehigh end. Turning to the bottom panel, again, both the square-root and the nonlinear CEV
models give a virtually same hyperbolic kurtosis function—shooting up at the lower end and
approaching three at the higher end. Note that the jump-diffusion model gives an extraor-
dinarily high kurtosis, ranging from 9 at the lower end to 44 at the higher end (outside and
above the picture range). Usually introducing jumps helps to increase the model skewness
and kurtosis, but an unusually high kurtosis of 9-44 must be caused by the restrictive jump
specification (constant jump rate ρt = ρ and uniform jump size J t ∼ U [−art, art]), which is
imposed to identify all the model parameters. The preferred quadratic model has a nonlinear
V-shaped kurtosis function for the short rate level between 0 and 8 percent and then mostly
a constant. In short, the quadratic variance model produces unique nonlinear conditional
skewness and kurtosis, which are dramatically different from all other candidate models.
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5 Conclusion
This paper proposes an Ito’s approach to generate the conditional moments for continuous
time Markov processes and gives a characterization of the class of admissible models. Theresulting conditional moment vector forms the basis of a natural GMM estimator. Monte
Carlo evidence suggests that such a moment generator and the related estimator behave rea-
sonably well for a benchmark square-root model. When applied to the empirical U.S. short
rate data, the procedure singles out the quadratic variance model as the only unrejected spec-
ification at the one percent level. The benchmark square-root model, the state-dependent
jump-diffusion process, and the nonlinear drift CEV model all fail in the GMM tests of overi-
dentifying restrictions. Further diagnostics suggests that the U-shaped conditional variance
and non trivial conditional skewness and kurtosis are important in modeling the short rate
dynamics in the univariate setting. One important extension is to estimate a multivariate
asset return model with possible quadratic volatility components.
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Table 1: Monte Carlo ExperimentThis table compares the finite sample performance of the GMM estimator proposed in thispaper with that of the MLE estimator provided by Durham and Gallant (2002). ∆ stands
for the discrete sampling interval and df for the degree of freedom of the implied non-centralchi-square distribution. The random sample size is chosen as 1000 and the number of MonteCarlo replicates is 512. Here we report the mean bias and root-mean-squared-error.
True Value MLE GMM MLE GMM
Mean Bias Mean Bias Root-MSE Root-MSE
Scenario (a), ∆ = 1/12, df = 5.33
κ = 0.50 0.0489 0.0828 0.1344 0.1473
θ = 0.06 0.0006 -0.0027 0.0080 0.0086
σ = 0.15 0.0002 -0.0028 0.0034 0.0046
Scenario (b), ∆ = 1/12, df = 2.48
κ = 0.50 0.0597 0.1036 0.1413 0.1697
θ = 0.06 -0.0003 -0.0052 0.0114 0.0118
σ = 0.22 0.0001 -0.0045 0.0054 0.0075
Scenario (c), ∆ = 1/12, df = 133.33
κ = 0.50 0.0438 -0.0042 0.1299 0.1221
θ = 0.06 0.0001 -0.0000 0.0016 0.0017
σ = 0.03 0.0002 -0.0003 0.0007 0.0008
Scenario (d), ∆ = 1/12, df = 4.27
κ = 0.40 0.0458 0.0892 0.1210 0.1446
θ = 0.06 0.0008 -0.0046 0.0102 0.0102
σ = 0.15 0.0001 -0.0029 0.0035 0.0048
Scenario (e), ∆ = 1/12, df = 53.33
κ = 5.00 0.0151 0.0169 0.4630 0.4580
θ = 0.06 0.0001 -0.0002 0.0008 0.0009
σ = 0.15 0.0000 -0.0040 0.0043 0.0059Scenario (f), ∆ = 2, df = 53.33
κ = 0.50 0.0013 0.0283 0.0430 0.0483
θ = 0.06 0.0004 -0.0022 0.0018 0.0029
σ = 0.15 0.0004 -0.0041 0.0056 0.0066
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Table 2: Summary Statistics of Three Month T-Bill RatesThe following table summarizes the weekly U.S. 3-month t-bill rates from January 1954 toJuly 2002 with a total of 2504 observations. The data is obtained from the public websiteof the Federal Reserve Bank of St. Louis.
Moments and Quantiles jth Order Autocorrelations
Mean 0.0542 ρ0 1.0000
Std. Dev. 0.0277 ρ1 0.9964
Skewness 1.1414 ρ2 0.9912
Kurtosis 4.8728 ρ3 0.9856
Minimum 0.0058 ρ4 0.9798
5%-qntl. 0.0171 ρ5 0.9734
25%-qntl. 0.0347 ρ6 0.9667
Medium 0.0504 ρ7 0.9600
75%-qntl. 0.0689 ρ8 0.9537
95%-qntl. 0.1045 ρ9 0.9477
Maximum 0.1676 ρ10 0.9418
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Table 3: Empirical Estimation ResultsThis table presents the main empirical results of the four model specifications discussed inSection 2 and estimated by the GMM estimator outlined in Section 3.
Square-Root Nonlinear CEV Jump-Diffusion Quadratic Variance
κ = 0.0020 0.0017 0.0005 0.0010
(0.0013) (0.0010) (0.0001) (0.0002)
θ = 0.0497 0.0458 0.0995 0.0669
(0.0131) (0.0128) (0.0186) (0.0016)
σ = 0.0062 0.0059 0.0031
(0.0002) (0.0002) (0.0012)
γ = 0.4825
(0.0028)
ρ = 0.0381
(0.0025)
a = 0.2196
(0.0265)σ0 = 0.0015
(0.0002)
σ1 = 0.0097
(0.0005)
σ2 = 0.0412
(0.0006)
Chi-Square = 49.3617 48.5714 31.6703 21.1222
d.o.f = 11 10 9 9
p-value = 0.0000 0.0000 0.0002 0.0121
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0 20 40 60 80 1000
20
40
60
80
100
Reference Curve
Rejection Curve
Nominal Level of Test
P e r c e n t a g e o f R e j e c t i o n
Scenario (a)
0 20 40 60 80 1000
20
40
60
80
100
Reference Curve
Rejection Curve
Nominal Level of Test
P e r c e n t a g e o f R e j e c t i o n
Scenario (b)
0 20 40 60 80 1000
20
40
60
80
100
Reference Curve
Rejection Curve
Nominal Level of Test
P e r c e n t a g e o f R e j e c t i o n
Scenario (c)
0 20 40 60 80 1000
20
40
60
80
100
Reference Curve
Rejection Curve
Nominal Level of Test
P e r c e n t a g e o f R e j e c t i o n
Scenario (d)
0 20 40 60 80 1000
20
40
60
80
100
Reference Curve
Rejection Curve
Nominal Level of Test
P e r c e n t a g e o f R e j e c t i o n
Scenario (e)
0 20 40 60 80 1000
20
40
60
80
100
Reference Curve
Rejection Curve
Nominal Level of Test
P e r c e n t a g e o f R e j e c t i o n
Scenario (f)
Figure 1: GMM Specification Test of Overidentifying Restrictions.
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−5 0 5
0
0.2
0.4
Scenario (a): κ
−5 0 5
0
0.2
0.4
Scenario (a): θ
−5 0 5
0
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Scenario (a): σ
−5 0 5
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Scenario (b): κ
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0
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Scenario (b): θ
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Scenario (b): σ
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Scenario (c): κ
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Scenario (c): θ
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Scenario (c): σ
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Scenario (d): κ
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Scenario (d): σ
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Scenario (e): θ
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Scenario (e): σ
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Scenario (f): κ
−5 0 5
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Scenario (f): θ
−5 0 5
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Scenario (f): σ
Figure 2: “- -” Normal (0,1) reference density; “—” t-test statistics.
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1960 1970 1980 1990 20000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18US 3−Month Treasury Bill Rate: 1954−2002
Figure 3: Time Series Plot of the Short Term Interest Rate.
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1x 10
−4 Conditional Mean
Short Rate Level
Zero as ReferenceSquare−RootNonlinear CEVJump−DiffusionQuadratic Variance
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
1
2
3
4
5
6x 10
−5 Conditional Variance
Short Rate Level
Square−RootNonlinear CEVJump−DiffusionQuadratic Variance
Figure 4: Conditional Mean and Variance.
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25Conditional Skewness
Short Rate Level
Zero as ReferenceSquare−RootNonlinear CEVJump−DiffusionQuadratic Variance
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.22.99
3
3.01
3.02
3.03
3.04
Conditional Kurtosis
Short Rate Level
Three as ReferenceSquare−RootNonlinear CEVJump−Diffusion (9−44)Quadratic Variance
Figure 5: Conditional Skewness and Kurtosis.